Cheng, Greiner, Kelly, Bell and Liu [Artificial Intelligence 137 (2002) 4390]
describe an algorithm for learning Bayesian networks that in a domain consisting
of n variables identifies the optimal solution using O(n4) calls to a
mutual-information oracle. This result relies on (1) the standard assumption that
the generative distribution is Markov and faithful to some directed acyclic graph
(DAG), and (2) a new assumption about the generative distribution that the
authors call monotone DAG faithfulness (MDF). The MDF assumption rests on an
intuitive connection between active paths in a Bayesian-network structure and the
mutual information among variables. The assumption states that the (conditional)
mutual information between a pair of variables is a monotonic function of the set
of active paths between those variables; the more active paths between the
variables the higher the mutual information. In this paper, we demonstrate the
unfortunate result that, for any realistic learning scenario, the monotone DAG
faithfulness assumption is incompatible with the faithfulness assumption.
Furthermore, for the class of Bayesian-network structures for which the two
assumptions are compatible, we can learn the optimal solution using standard
approaches that require only O(n2) calls to an independence oracle.